Biomechanical models based on the finite element method have already shown their potential in the simulation of the mechanical behaviour of cells. For instance, development of atherosclerosis is accelerated by damage of the endothelium, a monolayer of endothelial cells on the inner surface of arteries. Finite element models enable us to investigate mechanical factors not only at the level of the arterial wall but also at the level of individual cells. To achieve this, several finite element models of endothelial cells with different shapes are presented in this paper. Implementing the recently proposed bendotensegrity concept, these models consider the flexural behaviour of microtubules and incorporate also waviness of intermediate filaments. The suspended and adherent cell models are validated by comparison of their simulated force-deformation curves with experiments from the literature. The flat and dome cell models, mimicking natural cell shapes inside the endothelial layer, are then used to simulate their response in compression and shear which represent typical loads in a vascular wall. The models enable us to analyse the role of individual cytoskeletal components in the mechanical responses, as well as to quantify the nucleus deformation which is hypothesized to be the quantity decisive for mechanotransduction.

Institute of Solid Mechanics Mechatronics and Biomechanics Technicka 2896 2 61669 Brno Czech Republic

Zobrazit více v PubMed

Ross R. Atherosclerosis: an inflammatory disease. The New England Journal of Medicine. 1999;340(2):115–126. doi: 10.1056/NEJM199901143400207. PubMed DOI

Comerford A. Computational models of endothelial and nucleotide function. New Zealand: Doctoral thesis, University of Canterbury; 2007.

Sumpio B., Timothy Riley J., Dardik A. Cells in focus: endothelial cell. The International Journal of Biochemistry & Cell Biology. 2002;34(12):1508–1512. doi: 10.1016/S1357-2725(02)00075-4. PubMed DOI

Nieto A., Escribano J., Spill F., Garcia-Aznar J. M., Gomez-Benito M. J., Gomez-Benito M. J. Finite element simulation of the structural integrity of endothelial cell monolayers: a step for tumor cell extravasation. Engineering Fracture Mechanics. 2020;224, article 106718 doi: 10.1016/j.engfracmech.2019.106718. DOI

Oea R. D., King J. R. Continuum limits of pattern formation in hexagonal-cell monolayers. Journal of Mathematical Biology. 2012;64(3):579–610. PubMed

Ye M., Sanchez H. M., Hultz M., et al. Brain microvascular endothelial cells resist elongation due to curvature and shear stress. Scientific Reports. 2015;4:p. 4681. doi: 10.1038/srep04681. PubMed DOI PMC

Lebiš R. Czech Republic, Brno: PhD thesis, Mechanical Engineering, Brno University of Technology, Faculty of Mechanical Engineering; 2007. Computational modelling of mechanical behaviour of the cell.

Ingber D. Cellular tensegrity: defining new rules of biological design that govern the cytoskeleton. Journal of Cell Science. 1993;104:613–627. PubMed

Huang H., Dai C., Shen H., et al. Recent advances on the model, measurement technique, and application of single cell mechanics. International Journal of Molecular Sciences. 2020;21(17):p. 6248. doi: 10.3390/ijms21176248. PubMed DOI PMC

Nagayama K., Nagano Y., Sato M., Matsumoto T. Effect of actin filament distribution on tensile properties of smooth muscle cells obtained from rat thoracic aortas. Journal of Biomechanics. 2006;39(2):293–301. doi: 10.1016/j.jbiomech.2004.11.019. PubMed DOI

Ujihara Y., Nakamura M., Miyazaki H., Wada S. Contribution of actin filaments to the global compressive properties of fibroblasts. Journal of the Mechanical Behaviour of Biomedical Materials. 2012;14:192–198. doi: 10.1016/j.jmbbm.2012.05.006. PubMed DOI

Nguyen B. V., Wang Q., Kuiper N. J., Haj A. J., Thomas C. R., Zhang Z. Strain-dependent viscoelastic behaviour and rupture force of single chondrocytes and chondrons under compression. Biotechnology Letters. 2009;31(6):803–809. doi: 10.1007/s10529-009-9939-y. PubMed DOI

Pillarisetti A., Desai J., Ladjal H., Schiffmacher A., Ferreira A., Keefer C. Mechanical phenotyping of mouse embryonic stem cells: increase in stiffness with differentiation. Cellular Reprogramming. 2011;13(4):371–380. doi: 10.1089/cell.2011.0028. PubMed DOI

Kelsey M., Stroka M. K. G. Vascular endothelial cell mechanosensing: new insights gained from biomimetic microfluidic models. Seminars in Cell & Developmental Biology. 2017;71:106–117. PubMed

Kirill L., Yasaman N., Menglin S. Probing eukaryotic cell mechanics via mesoscopic simulations. PLoS Computational Biology. 2017;13:1–22. PubMed PMC

Daza R., González-Bermúdez B., Cruces J., et al. Comparison of cell mechanical measurements provided by Atomic Force Microscopy (AFM) and Micropipette Aspiration (MPA) Journal of the mechanical behavior of biomedical materials. 2019;95:103–115. PubMed

Mohammadkarim A., Mokhtari-Dizaji M., Kazemian A., et al. The mechanical characteristics of human endothelial cells in response to single ionizing radiation doses by using micropipette aspiration technique. Molecular & Cellular Biomechanics. 2019;16(4):275–287. doi: 10.32604/mcb.2019.06280. DOI

Barreto S., Clausen C., Perrault C., Fletcher D., Lacroix D. A multi-structural single cell model of force-induced interactions of cytoskeletal components. Biomaterials. 2013;34(26):6119–6126. doi: 10.1016/j.biomaterials.2013.04.022. PubMed DOI PMC

Ujihara Y., Nakamura M., Miyazaki H., Wada S. Proposed spring network cell model based on a minimum energy concept. Annals of Biomedical Engineering. 2010;38(4):1530–1538. doi: 10.1007/s10439-010-9930-8. PubMed DOI

Maurin B., Canadas P., Baudriller H., Montcourrier P., Bettache N. Mechanical model of cytoskeleton structuration during cell adhesion and spreading. Journal of Biomechanics. 2008;41(9):2036–2041. doi: 10.1016/j.jbiomech.2008.03.011. PubMed DOI

Deshpande V. S., McMeeking R. M., Evans A. G. A bio-chemo-mechanical model for cell contractility. Preceedings of national academy of science of the U.S.A. 2006;3(38):14015–14020. PubMed PMC

de R., Zemel A., Safran S. Dynamics of cell orientation. Nature Physics. 2007;3(9):655–659. doi: 10.1038/nphys680. DOI

Kaunas R., Hsu H. J. A kinematic model of stretch-induced stress fiber turnover and reorientation. Journal of Theoretical Biology. 2009;257(2):320–330. doi: 10.1016/j.jtbi.2008.11.024. PubMed DOI

Shen T., Shirinzadeh B., Zhong Y., Smith J., Pinskier J., Ghafarian M. Sensing and modelling mechanical response in large deformation indentation of adherent cell using atomic force microscopy. Sensors. 2020;20(6):p. 1764. doi: 10.3390/s20061764. PubMed DOI PMC

Kardas D., Nackenhorst U., Balzani D. Computational model for the cell-mechanical response of the osteocyte cytoskeleton based on self-stabilizing tensegrity structures. Biomechanics and Modelling in Mechanobiology. 2013;12(1):167–183. doi: 10.1007/s10237-012-0390-y. PubMed DOI

Alippi A., Bettucci A., Biagioni A. Nonlinear behaviour of cell tensegrity models. AIP Conference Proceedings. 2012;1433:329–332.

Bursa J., Lebis R., Holata J. Tensegrity finite element models of mechanical tests of individual cells. Technology and Health Care. 2012;20(2):135–150. doi: 10.3233/THC-2011-0663. PubMed DOI

Mehrbod M., Mofrad R. On the significance of microtubule flexural Behavior in cytoskeletal mechanics. PLoS One. 2011;6(10, article e25627) doi: 10.1371/journal.pone.0025627. PubMed DOI PMC

Bansod Y. D. Computational simulation of mechanical tests of isolated animal cells. Doctoral Thesis, Brno University of Technology; 2016. available from:

Bansod Y. D., Matsumoto T., Nagayama K., Bursa J. A finite element bendo-tensegrity model of eukaryotic cell. Journal of Biomechanical Engineering. 2018;140(10) doi: 10.1115/1.4040246. PubMed DOI

Rand R. Mechanical properties of the red cell membrane: II. Viscoelastic breakdown of the membrane. Biophysical Journal. 1964;4(4):303–316. doi: 10.1016/S0006-3495(64)86784-9. PubMed DOI PMC

Bachmann B. J., Giampietro C., Bayram A., et al. Honeycomb-structured metasurfaces for the adaptive nesting of endothelial cells under hemodynamic loads. Biomaterials science. 2018;6(10):2726–2737. doi: 10.1039/c8bm00660a. PubMed DOI

Pries A., Secomb T., Gaehtgens P. The endothelial surface layer. Pflügers Archiv-European Journal of Physiology. 2000;440(5):653–666. doi: 10.1007/s004240000307. PubMed DOI

Feletou M. The Endothelium: Part 1: Multiple Functions of the Endothelial Cells-Focus on Endothelium-Derived Vasoactive Mediators. San Rafael: Morgan & Claypool Life Sciences; 2011. PubMed

Carine M. Endothelial cell functions. Journal of Cellular Physiology. 2003;196:430–443. PubMed

Ingber D.  Tensegrity I. cell structure and hierarchical systems biology. Journal of cell science. 2003;116(7):1157–1173. doi: 10.1242/jcs.00359. PubMed DOI

Ingber D., Heidemann S., Lamoureux P., Buxbaum R. Opposing views on tensegrity as a strucutural framework for understanding cell mechanics. The American Physiological Society. 2000;89:1663–1678.

Janmey P., Euteneuer U., Traub P., Schliwa M. Viscoelastic properties of vimentin compared with other filamentous biopolymer networks. The Journal of Cell Biology. 1991;113(1):155–160. doi: 10.1083/jcb.113.1.155. PubMed DOI PMC

Wang N., Stamenović D. Contribution of intermediate filaments to cell stiffness, stiffening, and growth. American Journal of Physiology-Cell Physiology. 2000;279(1):C188–C194. PubMed

Stamenovic D., Wang N. Invited review. Engineering approaches to cytoskeletal mechanics. Journal of Applied Physiology. 2000;89(5):2085–2090. doi: 10.1152/jappl.2000.89.5.2085. PubMed DOI

Deguchi S., Ohashi T., Sato M. Evaluation of tension in actin bundle of endothelial cells based on preexisting strain and tensile properties measurements. Molecular & Cellular Biomechanics. 2005;2(3):125–133. PubMed

Kojima H., Ishijima A., Yanagida T. Direct measurement of stiffness of single actin filaments with and without tropomyosin by in vitro nanomanipulation. Proceedings of the National Academy of Sciences. 1994;91(26):12962–12966. doi: 10.1073/pnas.91.26.12962. PubMed DOI PMC

Centre for Bioengineering, Department of Mechanical Engineering, Trinity College Dublin, Ireland, McGarry J., Prendergast P. J. A three-dimensional finite element model of an adherent eukaryotic cell. eCM. 2004;7:27–34. doi: 10.22203/eCM.v007a03. PubMed DOI

Xue F., Lennon A. B., Mckayed K. K. Effect of membrane stiffness and cytoskeletal element density on mechanical stimuli within cells: an analysis of the consequences of ageing in cells. Computer Methods in Biomechanics and Biomedical Engineering. 2013;18:468–476. PubMed

Gittes F., Mickey B., Nettleton J., Howard J. Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape. Journal of Cell biology. 1993;120(4):923–934. doi: 10.1083/jcb.120.4.923. PubMed DOI PMC

Kiyoumarsioskouei A., Saidi M. S., Firoozabadi B. An endothelial cell model containing cytoskeletal components: suspension and adherent states. Journal of Biomedical Science and Engineering. 2012;5(12):737–742. doi: 10.4236/jbise.2012.512092. DOI

Caille N., Thoumine O., Tardy Y., Meister J. Contribution of the nucleus to the mechanical properties of endothelial cells. Journal of Biomechanics. 2002;35(2):177–187. doi: 10.1016/S0021-9290(01)00201-9. PubMed DOI

Stricker J., Falzone T., Gardel M. Mechanics of the F-actin cytoskeleton. Journal of Biomechanics. 2010;43(1):9–14. doi: 10.1016/j.jbiomech.2009.09.003. PubMed DOI PMC

Deguchi S., Yano N., Hashimoto K., Fukamachi H., Washio S., Tsujioka K. Assessment of the mechanical properties of the nucleus inside a spherical endothelial cell based on microtensile testing. Journal of Mechanics of Materials and Structures. 2007;2(6):1087–1102. doi: 10.2140/jomms.2007.2.1087. DOI

Jean R. P., Gray D. S., Spector A. A., Chen C. S. Characterization of the nuclear deformation caused by changes in endothelial cell shape. Journal of Biomechanical Engineering. 2004;126(5):552–558. doi: 10.1115/1.1800559. PubMed DOI

Unnikrishnan G., Unnikrishnan U., Reddy J. Constitutive material Modeling of cell: a micromechanics approach. Journal of Biomechanical Engineering. 2007;129(3):315–323. doi: 10.1115/1.2720908. PubMed DOI

Jean R. P., Chen C. S., Spector A. A. Finite-element analysis of the adhesion-cytoskeleton-nucleus mechanotransduction pathway during endothelial cell rounding: axisymmetric model. Journal of Biomechanical Engineering. 2005;127(4):594–600. doi: 10.1115/1.1933997. PubMed DOI

Ghaffari H., Saidi M. S., Firoozabadi B. Biomechanical analysis of actin cytoskeleton function based on a spring network cell model. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2017;231 doi: 10.1177/0954406216668546. DOI

Leipzig N. D., Athanasiou K. A. Static compression of single chondrocytes catabolically modifies single-cell gene expression. Biophysical Journal. 2008;94(6):2412–2422. doi: 10.1529/biophysj.107.114207. PubMed DOI PMC

Shieh A. C., Athanasiou K. A. Dynamic compression of single cells. Osteoarthritis and Cartilage. 2007;15(3):328–334. doi: 10.1016/j.joca.2006.08.013. PubMed DOI

Miyazaki H., Hasegawa Y., Hayashi K. Tensile properties of contractile and synthetic vascular smooth muscle cells. JSME International Journal Series C. 2002;45(4):870–879. doi: 10.1299/jsmec.45.870. DOI

Bursa J., Lebis R., Janicek P. FE models of stress-strain states in vascular smooth muscle cell. Technology and Health Care. 2006;14(4–5):311–320. doi: 10.3233/THC-2006-144-513. PubMed DOI

Lili W., Weiyi C. Modelling cell origami via a tensegrity model of the cytoskeleton in adherent cells. Applied Bionics and biomechanics. 2019;2019 doi: 10.1155/2019/8541303.8541303 PubMed DOI PMC

Wang L., Wang L., Xu L., Weiyi C. Finite element modeling of single-cell based on atomic force microscope indentation method. Computational and Mathematical Methods in Medicine. 2019;2019 doi: 10.1155/2019/789506.7895061 PubMed DOI PMC

Katti D., Katti K. Cancer cell mechanics with altered cytoskeletal behavior and substrate effects: a 3D finite element modeling study. Journal of the Mechanical Behavior of Biomedical Materials. 2017;76:125–134. doi: 10.1016/j.jmbbm.2017.05.030. PubMed DOI

Mofrad M. R. K., Kamm R. D. Cytoskeletal mechanics: models and measurements. Cambridge University Press; 2009. DOI

Nolting J.-F., Möbius W., Köster S. Mechanics of individual keratin bundles in living cells. Biophysical Journal. 2014;107(11):2693–2699. doi: 10.1016/j.bpj.2014.10.039. PubMed DOI PMC

Huang H., Kamm R. D., So P. T., Lee R. T. Receptor-based differences in human aortic smooth muscle cell membrane stiffness. Hypertension. 2001;38(5):1158–1161. doi: 10.1161/hy1101.096456. PubMed DOI

Long-term zinc treatment alters the mechanical properties and metabolism of prostate cancer cells